An improved hippopotamus optimization algorithm based on adaptive development and solution diversity enhancement

The workflow of the HO

This section introduces the fundamental HO. HO is inspired by the unique behaviors of hippopotamuses, incorporating their natural traits to address optimization challenges. By understanding the foundational principles and mechanisms of HO, we establish a basis for the subsequent improvements proposed in this study.

$X_{P1}$ represents the updated individual position in update phase 1. This position is adjusted randomly based on the distance from the dominant hippopotamus (best solution) to explore the solution space.

(1) $$X_{P1}(i,:)=X(i,:)+rand(0,1)\times (X_{best}-I_1\times X(i,:)),i=1,2,..,\lfloor \frac{N}{2}\rfloor$$where the updated position $X_{P1}(i,:)$ of the $i$-th individual in the first half of the population. $X(i,:)$ is the current position of the $i$-th individual. $X_{best}$ is the distance to the current global best position in the population. N is total number of individuals in the population. $m$ is number of dimensions in the solution space. $rand$ is a random factor that influences the magnitude of the position update. $I_1$ is a weighting factor that controls the influence of the individual’s current position.

$X_{P2}$ represents the updated individual position in update phase 1. When the temperature parameter T is high, this position is adjusted based on the dominant hippopotamus and the mean of a random group. When T is low, it is adjusted based on the difference between the mean and the dominant hippopotamus or randomly generated within the bounds.

(2) $$X_{P2M}(i,:)=\{\begin{array}{cc}X(i,:)+B\times (X_{mean}-X_{best})\hfill & ,\mathrm{i}\mathrm{f}rand(0,1)>0.5\hfill \\ lb+rand(0,1)\times (ub-lb)\hfill & ,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}$$where B is a random scaling factor that influences the magnitude of the position update. $rand(0,1)$ is random numbers used to decide which movement rule to apply. $lb$, $ub$ is the lower and upper bounds of the search space. $X_{mean}$ is the mean position of a group of individuals, which serves as a reference for the update. $X_{best}$ is the distance to the best-performing (dominant) hippopotamus in the population. If $rand(0,1)>0.5$, the position is updated with respect to the mean group position $X_{mean}$ and the distance to the best-performing individual $X_{best}$. Otherwise, a random position is generated within the bounds of the search space.

(3) $$X_{P2}(i,:)=\{\begin{array}{cc}X(i,:)+A\times (X_{best}-I_2\times X_{mean}),\hfill & \mathrm{i}\mathrm{f}\mathrm{T}>0.6\hfill \\ X_{P2M},\hfill & \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\hfill \end{array}$$where T is adaptive parameter that determines whether strong or weak exploitation is used. $I_2$ is a weight factor controlling the balance between exploration and exploitation. A is a random movement factor for controlling the exploration step size.

$X_{P3}$ represents the updated individual position in the defense phase, adjusted based on the distance between the hippopotamus and the predator as well as the Levy flight strategy.

(4) $$\overrightarrow{P}=lb+rand(1,D)\times (ub-lb)$$ $lb$ and $ub$ are the lower and upper bounds of the search space. The predator’s position $\overrightarrow{P}$ is initialized randomly between the lower bound $lb$ and upper bound $ub$ in each dimension.

(5) $$Levy(u,v,\beta )=\frac{u}{|v{|}^{1/\beta }}$$where $u\sim \mathcal{N}(0,{\sigma }_u^2)$ and $v\sim \mathcal{N}(0,1)$ are random variables drawn from normal distributions. This equation provides Levy that follow a heavy-tailed distribution, which is often used in stochastic optimization algorithms to allow both small and occasional large jumps for efficient exploration of the solution space.

(6) $${\sigma }_u={\left(\frac{\mathrm{\Gamma }(1+\beta )\cdot sin(\frac{\pi \beta }{2})}{\mathrm{\Gamma }(\frac{1+\beta }{2})\cdot \beta \cdot 2^{(\beta -1)/2}}\right)}^{1/\beta }$$where $\mathrm{\Gamma }$ denotes the gamma function. $\beta$ is the power-law index, where $1<\beta <2$.

(7) $$\mathrm{D}\mathrm{L}=\left|\overrightarrow{\mathrm{P}}-\mathrm{X}(\mathrm{i},:)\right|$$

(8) $$@X_{P3}(i,:)=\{\begin{array}{cc}RL(i,:)\times \overrightarrow{P}+\left(\frac{b}{c-d\times cos(l)}\right)\times \left(\frac{1}{DL}\right)\hfill & ,\mathrm{i}\mathrm{f}f(i)>f(\overrightarrow{P})\hfill \\ RL(i,:)\times \overrightarrow{P}+\left(\frac{b}{c-d\times cos(l)}\right)\times \left(\frac{1}{2\times \mathrm{D}\mathrm{L}+rand(1,D)}\right)\hfill & ,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}$$where $b$, $c$, $d$ are random scaling coefficients sampled from uniform distributions. $rand$ is a random angle within the range $[1,D]$.

Levy flight RL is a common random walk model used to simulate animal foraging paths, and in this case, it’s represented as: $RL=0.05\times Levy(S,D,\beta )$. S is the number of search agents.D is the dimensionality. $\beta =1.5$ is the Levy exponent parameter.

$X_{P4}$ represents the updated individual position in the escape phase, where positions are adjusted with local random perturbation within the range between the lower and upper bounds.

(9) $$X_{P4}(i,:)=X(i,:)+rand(0,1)\times (l{b}^{local}+D\times (u{b}^{local}-l{b}^{local})),i=1,2,...,N$$where $u{b}^{local}=ub/t$ and $l{b}^{local}=lb/t$ represent the adaptive local lower and upper bounds for the hippopotamus positions, adjusting over iterations to focus the search progressively. $t$ represents the current iteration number in the optimization algorithm.

Detailed description of the three main phases

HO consists of three main stages: exploration, defense, and exploitation. Each phase plays a crucial role in balancing the search process between exploring the solution space broadly and refining around promising areas.

Limitations of HO in solving complex optimization problems

Although the HO demonstrates strong search capabilities in both the exploration and exploitation phases, it may encounter the following limitations when addressing complex, high-dimensional optimization problems.

Despite the inclusion of the escape mechanism, the algorithm remains susceptible to local optima, particularly in complex problems, which can lead to performance degradation. Additionally, in large-scale problems, the convergence speed of HO may not be ideal. The performance of the algorithm is also highly sensitive to parameter settings, such as the balance factor between exploration and exploitation, further impacting its overall effectiveness.

Overview of the improvement strategies

The improvement strategies proposed in this article include three key components. First, chaotic map initialization utilizes logistic chaotic mapping to generate a diverse initial population, thereby enriching the solution space. Second, an adaptive exploitation module dynamically adjusts the intensity of exploitation to enhance search capability across different iteration stages. Third, a solution diversity enhancement mechanism is applied after the exploitation and escape phases, incorporating Gaussian mutation and chaotic perturbation to maintain diversity within the population and effectively prevent premature convergence.

Chaotic map initialization strategy and its role

The diversity of the initial population has a significant impact on the global search performance of the algorithm. This article introduces a population initialization method based on the Logistic chaotic map in the standard HO to increase the diversity of the initial solutions and improve the exploration capability in the early stages of the search. The Logistic chaotic map is defined by the following Eq. (10).

(10) $$X_{n+1}=r\times X_n\times (1-X_n)$$where $r$ is the chaotic factor, typically set to 4. The initial solutions generated using chaotic mapping can more evenly cover the search space, enhancing the algorithm’s global search capability.

Design and implementation of the adaptive exploitation mechanism

In the standard HO, the exploitation intensity is fixed, and as the number of iterations increases, the search capability gradually decreases. To address this issue, this article introduces an adaptive exploitation strategy, allowing the exploitation intensity to dynamically adjust based on the iteration count. In the early stages of the algorithm, strong exploitation is used to quickly converge, while in the later stages, fine-tuned exploitation increases the likelihood of finding the global optimum.

The inertia weight $w$ in the algorithm is dynamically updated each iteration to balance exploration and exploitation. It starts at a higher value (0.9) and gradually decreases to a minimum value (0.4) as iterations progress, following the Eq. (11).

(11) $$w=w_{min}+(0.9-w_{min})\times (1-t/t_{max})$$where $t$ is the current iteration and $t_{max}$ is the total number of iterations. This adaptive decay allows the search process to initially explore widely, then converge more precisely in later stages. (12) $$X_{P1}(i,:)=X(i,:)+w\times rand(0,1)\times (X_{best}-I_1\times X(i,:)),i=1,2,..,\lfloor \frac{N}{2}\rfloor$$

(13) $$X_{P4}(i,:)=X(i,:)+w\times rand(0,1)\times (l{b}^{local}+D\times (u{b}^{local}-l{b}^{local})),i=1,2,...,N.$$

The adaptive exploitation strategy effectively enhances the search capability at different stages of the iteration.

Mutation-based solution diversity enhancement mechanism

To prevent the algorithm from getting stuck in local optima, this article introduces mutation operations to enhance solution diversity. Specifically, Gaussian mutation is applied to solutions during the exploitation and escape phases to increase the population’s ability to jump out of local regions. The formula for Gaussian mutation is as follows Eq. (14).

(14) $$r_{mutation}=0.1+0.9\times (1-t/t_{max})$$where $r_{mutation}$ is the mutation strength parameter that controls the mutation amplitude. This operation increases the randomness of the solutions and enhances the algorithm’s ability to escape from local optima.

(15) $$A=r_{mutation}\times (2\times rand(1,D)-1)$$

(16) $$B=r_{mutation}\times rand(1,D).$$

This enhancement introduces multiple update strategies for the positions $X_{P2M}$ (Eq. (2)) and $X_{P2}$ (Eq. (3)), incorporating dynamic mutation strategies A and B to regulate the update process. During each iteration, the algorithm uses a combination of inertia-weighted and dynamically adjusted mutation strategies to improve exploration capabilities.

Pseudocode for the improved algorithm

The improvements in the proposed algorithm are threefold, each designed to address specific limitations. First, the inertia weight is gradually reduced to balance exploration and exploitation, as maintaining a fixed inertia weight can lead to premature convergence or insufficient exploration. This dynamic adjustment enhances the algorithm’s performance in the exploration phase. Second, an adaptive mutation rate adjustment is introduced to address the challenge of stagnation in later stages. By dynamically tuning the mutation rate based on the iteration count, the algorithm achieves finer searches in later stages, improving overall exploration effectiveness. Finally, in the predator escape phase, local boundary updates and position adjustments are incorporated to overcome the problem of slow convergence in complex search spaces. This adjustment improves the algorithm’s global convergence and escape capabilities during the exploitation phase, allowing it to better explore optimal solutions.

Figure 1 presents the algorithmic flowchart of the improved hippopotamus optimization algorithm, while Algorithm 2 provides the pseudocode for the enhanced HO.

In the Fig. 1, green rectangles represent Phase 1, the adaptive exploitation phase; yellow rectangles indicate Phase 2, the defense phase; and blue rectangles denote Phase 3, the escape phase.

Experimental setup: benchmark functions and parameter configuration

In this study, we compared the effectiveness of the improved IHO with nine classical metaheuristic algorithms, including HO, PSO, sine cosine algorithm (SCA), firefly algorithm (FA), TLBO, Evolution Strategy with Covariance Matrix Adaptation (CMA-ES), moth flame optimization (MFO), arithmetic optimization algorithm (AOA), Invasive Weed Optimization (IWO), Improved Sand Cat Swarm Optimization (ISCSO), penguin jump algorithm (PGJA) and WOA. The control parameters for these algorithms were carefully adjusted according to the specific descriptions provided in Table 1. This section presents the simulation studies of IHO on various complex optimization problems. The effectiveness of IHO in obtaining optimal solutions was evaluated through a comprehensive set of 68 standard benchmark functions. These benchmark functions include unconstrained problems, high-dimensional problems, multi-modal problems and engineering optimization problems. To further evaluate the performance of the algorithms on F1 to F23 (CEC05), 30 independent runs were performed, evaluate the performance of the algorithms on F1 to F30 (CEC17), 50 independent runs were performed, evaluate the performance of the algorithms on F1 to F12 (CEC22), 50 independent runs were performed, evaluate the performance of the algorithms on engineering optimization problems, 30 independent runs were performed. The population size for IHO was set to 24 members in AOA and TLBO, while other algorithms used 30 or 60 members, with the maximum number of iterations set to 1,000 on CEC05. The optimization results are presented using five comprehensive statistical metrics: mean, best, worst, standard deviation, and median. Among these, the mean index was particularly used as a key ranking parameter to evaluate the effectiveness of the metaheuristic algorithms on each benchmark function.

This study evaluated 23 functions (CEC05), among which F1–F7 are unimodal (UM) functions, F8–F13 are high-dimensional multimodal (HM) functions, and F14–F23 include fixed-dimensional multimodal (FM) and multimodal (MM) functions.

Specifically, the simulation environment was as follows: Windows 10, Intel Xeon CPU E5-1660 3.0 GHz, 128 GB memory.

Algorithm performance comparison

This section provides a detailed comparison of the performance of IHO against the standard HO and other widely used algorithms. The experimental results demonstrate the superior performance of IHO in terms of solution accuracy, convergence speed, and global search capability, particularly for multi-modal functions. Tables 2–4 shows the evaluation results of the benchmark functions, while Figs. 2–4 illustrates the convergence behavior of the five most effective algorithms when optimizing F1–F23 (CEC05).

Figures 2–4 illustrates the convergence curves of the five most effective algorithms during the optimization process for F1–F23. This evaluation, as presented in Table 2, aims to assess the algorithms’ local search capabilities on eight distinct unimodal functions (F1–F8). IHO achieved global optima on F1–F4 and F5–F6, making it the only algorithm among the nine evaluated to reach this level of performance. Its performance on F4 significantly surpassed that of the other algorithms. In the highly competitive F6 test, IHO reached global optima alongside four other algorithms. Additionally, IHO demonstrated clear superiority on F7 and F8. For F1–F4 and F6, IHO consistently converged with a standard deviation of zero. For F7, the standard deviation was 5.21E−05, and for F5, it was 3.57E−2. Compared to the other algorithms, IHO exhibited the lowest standard deviation, indicating remarkable stability.

Figures 5–7 displays the box plots of the optimal values of the objective function obtained from 30 independent runs on F1–F23 (CEC05), utilizing IHO and five other algorithms.

Table 3 presents the results for HM functions on F9–F16, tested using different algorithms. These functions were chosen to evaluate the global search capabilities of the algorithms. IHO significantly outperformed all other algorithms on F12 and F13. In F9, IHO achieved global optima along with HO and AOA, demonstrating superior performance over the other algorithms. On F10, IHO performed comparably to HO and surpassed all other algorithms. For F11, IHO converged to the global optimum alongside HO and TLBO, showcasing exceptional performance. IHO also outperformed all other algorithms on F12, and in F13, IHO ranked first. In F16, IHO exhibited a notably lower standard deviation than some algorithms. For F13, the standard deviation was 5.55E−3, outperforming all other algorithms. These findings indicate that IHO demonstrates robust resilience in effectively handling these functions (refer to Fig. 3).

Furthermore, an assessment was conducted to examine the algorithm’s ability to balance exploration and exploitation during searches on F17–F23, with results recorded in Table 4. IHO exhibited the best performance on F17–F23, achieving a significantly lower standard deviation, particularly for F21–F23. These results suggest that IHO, characterized by a strong capacity to balance exploration and exploitation, exhibits outstanding performance when addressing FM and MM functions.

In benchmarked tests for CEC17 and CEC22, IHO was compared against four algorithms, including ISCSO, PGJA, SCA, and WOA. We run each problem 51 times independently. Comparison results between IHO and other algorithms are shown in Tables 5 to 15.